LMFDB - Newform orbit 9072.2.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.4402847137$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 2 q^{5} + q^{7}+O(q^{10}) $

$ q + 2 q^{5} + q^{7} - q^{11} - 6 q^{13} - 5 q^{17} + 7 q^{19} - 4 q^{23} - q^{25} - 4 q^{29} + 6 q^{31} + 2 q^{35} + 2 q^{37} + 3 q^{41} + q^{43} + q^{49} + 12 q^{53} - 2 q^{55} + 7 q^{59} - 12 q^{61} - 12 q^{65} - 13 q^{67} + 8 q^{71} + q^{73} - q^{77} + 6 q^{79} - 16 q^{83} - 10 q^{85} - 6 q^{89} - 6 q^{91} + 14 q^{95} - 5 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

2.00000

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9072.2.a.t		1
3.b	odd	2	1		9072.2.a.f		1
4.b	odd	2	1		1134.2.a.c		1
9.c	even	3	2		1008.2.r.a		2
9.d	odd	6	2		3024.2.r.c		2
12.b	even	2	1		1134.2.a.f		1
28.d	even	2	1		7938.2.a.e		1
36.f	odd	6	2		126.2.f.b	✓	2
36.h	even	6	2		378.2.f.b		2
84.h	odd	2	1		7938.2.a.bb		1
252.n	even	6	2		882.2.h.g		2
252.o	even	6	2		2646.2.h.b		2
252.r	odd	6	2		2646.2.e.h		2
252.s	odd	6	2		2646.2.f.b		2
252.u	odd	6	2		882.2.e.a		2
252.bb	even	6	2		2646.2.e.i		2
252.bi	even	6	2		882.2.f.f		2
252.bj	even	6	2		882.2.e.e		2
252.bl	odd	6	2		882.2.h.h		2
252.bn	odd	6	2		2646.2.h.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.f.b	✓	2	36.f	odd	6	2
378.2.f.b		2	36.h	even	6	2
882.2.e.a		2	252.u	odd	6	2
882.2.e.e		2	252.bj	even	6	2
882.2.f.f		2	252.bi	even	6	2
882.2.h.g		2	252.n	even	6	2
882.2.h.h		2	252.bl	odd	6	2
1008.2.r.a		2	9.c	even	3	2
1134.2.a.c		1	4.b	odd	2	1
1134.2.a.f		1	12.b	even	2	1
2646.2.e.h		2	252.r	odd	6	2
2646.2.e.i		2	252.bb	even	6	2
2646.2.f.b		2	252.s	odd	6	2
2646.2.h.b		2	252.o	even	6	2
2646.2.h.c		2	252.bn	odd	6	2
3024.2.r.c		2	9.d	odd	6	2
7938.2.a.e		1	28.d	even	2	1
7938.2.a.bb		1	84.h	odd	2	1
9072.2.a.f		1	3.b	odd	2	1
9072.2.a.t		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9072))$:

$ T_{5} - 2 $

$ T_{11} + 1 $

$ T_{13} + 6 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T - 2 $ T - 2
$7$	$ T - 1 $ T - 1
$11$	$ T + 1 $ T + 1
$13$	$ T + 6 $ T + 6
$17$	$ T + 5 $ T + 5
$19$	$ T - 7 $ T - 7
$23$	$ T + 4 $ T + 4
$29$	$ T + 4 $ T + 4
$31$	$ T - 6 $ T - 6
$37$	$ T - 2 $ T - 2
$41$	$ T - 3 $ T - 3
$43$	$ T - 1 $ T - 1
$47$	$ T $ T
$53$	$ T - 12 $ T - 12
$59$	$ T - 7 $ T - 7
$61$	$ T + 12 $ T + 12
$67$	$ T + 13 $ T + 13
$71$	$ T - 8 $ T - 8
$73$	$ T - 1 $ T - 1
$79$	$ T - 6 $ T - 6
$83$	$ T + 16 $ T + 16
$89$	$ T + 6 $ T + 6
$97$	$ T + 5 $ T + 5
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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